As a useful insight, the inequality of Eq. (4), which guarantees the existence of HFS entanglement, can be expressed via the Manhattan-norm coherence, \(D=\sum _{{i\ne j}} |\rho _{i,j}|\) , also referred to as the \(l_1\) -norm coherence [1]}. For the HFS states of the H atom at thermal equilibrium with a temperature \(\tau \) , this coherence is given by
\(D= \frac{\sinh [2\beta \mathcal {A} \sqrt{1+\xi ^2}]}{\mathcal {G}\sqrt{1+\xi ^2}}.\) 
